Estimate sin(3) using a tangent line approximation at π.
3 is close to π ≅ 3.14, so it makes sense to find the tangent line to f(x) = sin(x) at 3 and use that to approximate sin(3):
We'll give a freebie right now that f'(π) = -1. We know f(π) = sin(π) = 0. Now we build the tangent line equation:
y = mx + b
y = (-1)x + b
0 = (-1)(π) + b
b = π
y = (-1)x + π
Now we find the y-value on the tangent line when x = 3:
(-1)(3) + π ≅ 0.14.
This is close to sin(3).
Yes, it's possible to plug "sin(3)" into a calculator and find an answer. A lot of people don't realize the calculator is doing essentially the same thing we are: it's estimating. It's doing something more like a tangent line approximation on a caffeine high, and it's doing it faster than we can, but it's still estimating.
If the end of the world comes and all the calculators are destroyed, we'll still be able to calculate the sin of 3. We hope that's a reassuring thought.